In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as habitats for the ends of open strings and can be conceived of as submanifolds (of the target manifold in a sigma model), possibly augmented with a vector bundle, or a sheaf of [somethings], or maybe some other kind of label/data. (Corrections welcome.)

In the hopes of tightening my grasp on the concept, here are some of the questions that have been nagging me during my reading.

What specifically is the definition of a D-brane, say in the context of a topological field theory? (Or what are the most promising provisional definitions?) What references are most accessible to a mathematical audience?

What picture should I have in my head when an author talks about "the moduli space of D-branes"?

What is the idea behind the "dynamics of D-branes" that researchers sometimes talk about? (Perhaps when I understand better how to think about these gadgets, it will be easier to conceive of how they should change over time.)

What goes into verifying (or at least asserting/conjecturing) that the elements of twisted K-theory classify "D-brane charges"?

A small nitpick, but one which I think it's important. D-branes are not the submanifolds that the open strings end on. That's like saying that a (point) particle is a point in the spacetime. There are D-submanifolds on which open strings end, and then there are the D-branes which are physical objects occupying such D-submanifolds.
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José Figueroa-O'FarrillOct 23 '10 at 0:05

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My earlier comment notwithstanding, my gut feeling is that this question is a bit of a "fishing expedition". You are basically asking people to write a wikipedia-like article on D-branes and that's a lot of work. Your questions are not uninteresting, but answering them in any serious way, requires more effort than it seems you have spent on the question. I'm not going to vote to close yet, but I would suggest you try to make the question more precise.
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José Figueroa-O'FarrillOct 23 '10 at 0:09

An introduction to D-branes with a somewhat more mathematical slant was written by Kapustin and Orlov, published in Russian Math. Surveys 59, 2004, 907-940. The preprint version can be found in the arXiv arxiv.org/abs/math/0308173.
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LaieOct 23 '10 at 0:13

3 Answers
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There is an abstract algebraic formulation of QFT: this says that an $n$-dimensional QFT is a consistent assignment of spaces of states and of maps between them to $n$-dimensional cobordisms .

If one allows cobordisms with boundary here, one speaks of open-closed QFT . A D-brane in this context is the type of data assigned by the QFT to these boundaries.

There is also a geometric aspect to this: many abstractly defined QFTs are imagined to be "sigma-models". They are supposed to be induces by a process called "quantization" from a functional (the "action") on a space of maps $\Sigma \to X$ from a cobordism $\Sigma$ into a smooth manifold $X$ equipped with extra geometric data (such as metric, connections, etc.)

Under this correspondence one may ask which abstract algebraic properties of the the QFT derive from which geometric aspects of these "background structures". One finds that the data that the QFT assignes to boundaries comes from geometric data on $X$ that tends to look like submanifolds with their own geometric data on them (but may be considerably more general than that!). If so, this geomwetric data on $X$ is called a D-brane of the sigma-model.

There are many instances of this that are understood at the rough level at which quantum field theory was understood in the 20th century. One special case that is by now under fairly complete mathematical control and which sereves as a good guide to the general concept of D-branes is what is called "2d rational CFT" .

There is a complete mathematical classification of 2d rational CFTs in their abstract algebraic form: they are given by special symmetric Frobenius algebra objects internal to a modular tensor category of representation of a vertex operator algebra.

Under this classification theorem, the boundary data = D-branes in the algebraic formulation can be proven to be precisely modules over this Frobenius algebra object.

In special nice cases one understands where these come from geometrically. The notable example is the Wess-Zumino-Witten model, where the target space is a group manifold. Here one finds that in the simplest case the geometric data corresponding to these D-branes are submanifolds given by conjugacy classes, and carrying twisted vector bundles. More generally, though, the D-branes are given by cocycles in the twisted differential K-theory of the group. So the identification "D-brane = submanifold" is too naive, in general. The correct identification is:

I'm going to attempt a short, partial answer written for pure mathematicians.

The word "brane" in high-energy physics means "submanifold". The word is short for "membrane". More precisely, it means a submanifold of space that moves in time. A $p$-dimensional brane in space is therefore also a $p+1$-dimensional brane in spacetime.

The letter "D" is for "Dirichlet". In classical differential equations, a Dirichlet boundary condition is a condition that some of the function of the equation are held constant at the boundary of the domain. Geometrically, if $M$ is a manifold and $B \subseteq M$ is a "brane", then $B$ is used as a D-brane if a map $f:\Sigma \to M$ is (a) is mentioned in a differential equation, and (b) is required to satisfy the Dirichlet condition $f(\partial \Sigma) \subseteq D$. For example, if $\Sigma$ is a surface and $f$ is meant to satisfy the minimal surface equation, then the condition $f(\partial \Sigma) \subseteq D$ asks for a minimal surface whose boundary is a loop in $D$ (say in some homotopy class).

This picture is now subject to two reinterpretations. First, the quantum reinterpretation. Instead of demanding that $f$ satisfy its equation, we assume some Lagrangian functional $L$, and we study the formal integral of $\exp(iL(f))$ over all choices of $f$. For instance, instead of imposing the minimal surface equation, we can let $L$ be the area functional of $f(\Sigma)$, or the energy functional. This is often a non-rigorous integral (a Feynman path integral), but nonetheless in favorable cases it appears to be a would-be-rigorous integral and it can be studied. Looking at the space of all maps $f:\Sigma \to M$ for a manifold $M$, with or without a Dirichlet boundary condition, is called a "sigma model" in quantum field theory.

The second reinterpretation is that of string theory. Assume that $\Sigma$ is two-dimensional and that the sigma model is a conformal quantum field theory. Instead of just viewing this model as a quantum field theory, it is viewed as the perturbative expansion (as a power series in the genus of $\Sigma$) of some dynamical theory of $M$ itself, and maybe $M$ and some D-branes in $M$. In order for this second reinterpration to be viable, both $M$ and D-branes in $M$, and the other rules of the sigma model, have to satisfy a number of special conditions. (For instance, without extra decorations such as D-branes, $M$ must satisfy the Einstein equation.) These special conditions are sometimes part of the de facto definition of a D-brane.